Very long story short: it's a cut-down replica of the Bristol High Cross. The original used to stand in the centre of Bristol, erected in 1373 to commemorate the granting of a charter by Edward III to make Bristol a county, separate from Somerset and Gloucestershire, and now stands in the Stourhead Estate. This replica was made in 1851, and originally sited on College Green.
Facing us is Charles I, who used to look down Broad Street, and whose beard appears to have fallen off. On the right is Elizabeth I (Corn Street); on the left, Henry VI (Wine Street), and out of sight around the back is James I (High Street.) Directions and ID courtesy the interesting Wikipedia article.
I found out during later research into the Hughes family (including [James] Donald Hughes, who lived at 23 Berkeley Square) that it was Ellard Hughes, Donald's younger brother, who saved the upper part of the cross from the scrapheap and had it re-erected here. His father, Walter William Hughes, was an estate agent and general man of property with strong College Green connections, including being involved in the purchasing of land for the Council House, now City Hall1. I think it was during that work that the cross was removed from College Green.
This Tweet from Nick Howes shows the full cross standing in the centre of College Green in an aerial photo from 1931-2.
1 Source: A College Green Man, Article by William R Hughes on the many connections his father Walter William Hughes (1833 - 1909)
had with life round College Green, Bristol, Bristol Archives 30508, Deeds and documents of the Hughes family, estate agents of Bristol, 1791-1910.